Detection of Cardiovascular Anomalies: Hybrid Systems Approach

Proceedings of the 4th IFAC Conference on Analysis and Design of Hybrid Systems (ADHS 12) June 6-8, 2012. Eindhoven, The Netherlands

Detection of Cardiovascular Anomalies: Hybrid Systems Approach Fernando Diaz Ledezma* Taous Meriem Laleg-Kirati** King Abdullah University of Science and Technology (KAUST) 4700 King Abdullah University if Science and Technology Thuwal 23955-6900, Kingdom of Saudi Arabia e-mail: fernando. diaz@ kaust. edu. sa * e-mail: taousmeriem. laleg@ kaust. edu. sa ** Abstract: In this paper, we propose a hybrid interpretation of the cardiovascular system. Based on a model proposed by Simaan et al. (2009), we study the problem of detecting cardiovascular anomalies that can be caused by variations in some physiological parameters, using an observerbased approach. We present the first numerical results obtained. 1. INTRODUCTION Hybrid system modeling has been used in the recent years to represent the dynamics of biological systems. In a hybrid dynamical system the state evolves in continuous time as well as in discrete modes influenced by internal conditions of the model or external events. In biological systems, discrete behaviors might origin from unexpected changes in normal performance, e.g., a transition from a healthy to an abnormal condition. Simplifications, model assumptions, and/or modeled (and ignored) nonlinearities, can be represented by sudden changes in the state. For instance, in systems governed by a few molecules there are often observed inherently discrete processes, Belta et al. (2004). Hybrid systems are important for modeling nonlinearities in the dynamics of biological and medical systems. Discontinuities, as threshold-triggered firing in neurons, onoff switching of gene expression by a transcription factor, division in cells and certain types of chronotherapy for prostate cancer are common examples of these nonlinearities, Aihara and Suzuki (2010). Lincoln and Tiwari (2004) studied the hybrid conception of the glucose metabolism in humans. In this analysis, the body is divided into compartments that have their own glucose dynamics and represent the state variables. These dynamics include nonlinear functions and/or abrupt changes in the behavior corresponding to changes in one or many of the variables. The change in the concentration (when it rises above or falls below certain level) of glucose will redefine the state equations. The usage of hybrid systems is then suitable for biological systems as they provide the means to model their behavior with good approximations and simplifications. Another cause of discrete behavior in biological systems is the existence of processes that can be treated as naturally discrete phenomena, Aihara and Suzuki (2010). On the medical point of view, the appearance of discrete changes can be considered as consequences of failure in an organ. In such a situation there will be a change in the normal conditions. This transition can as well be captured in a discrete way. 978-3-902823-00-7/12/$20.00 © 2012 IFAC

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In this paper, we are interested in the detection of anomalies in the cardiovascular system (CVS) using fault detection and isolation (FDI) tools. The CVS is one of the most fascinating but most complex physiological systems. Cardiovascular diseases (CVD’s) constitute one of the most important causes of mortality in the world. An estimate of 17.3 million people died in 2008 from CVD’s, WHO (2012). Therefore, many studies have been devoted to modeling the CVS in order to well understand its behavior and to find new reliable diagnosis techniques. Hybrid properties appear naturally in the CVS thanks to the presence of valves which depending on their state (close or open) divide the cardiac cycle into four phases. Our objective is to use the properties of hybrid systems to describe this complex system in such a way that enables us to detect some anomalies. Faults that can occur in a hybrid system are either related with the current mode behavior or the trajectory of the discrete evolution. We are interested in modelbased methods for FDI. In these methods the actual signal measured from the system and the expected signal given by a mathematical model of the system are compared to generate residuals, i.e. errors, which, depending on their value, might indicate the appearance of faults. Our proposal is to use a model of the CVS that is able to describe the basic elastic properties of the arteries together with the relevant pressures in the system. As it will be further explained, this model can be ideally interpreted as a hybrid system. We will then extend the model in such a way that allows us to account for possible anomalies in the system. Later, an observer-based method will be used to detect the occurrence of anomalies. In the next sections, we will introduce the CVS and describe the existing approaches to model this complex system. Then we will introduce a simplified model of this system proposed by Simaan et al. (2009) and give a hybrid interpretation of it. In section 4, we will tackle the problem of detecting anomalies using an observer-based approach. We will relate some diseases to variations in physiological parameters, well described by the model. Section 5 will 10.3182/20120606-3-NL-3011.00059

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present preliminary results. Finally we will summarize and discuss the results. 2. BLOOD FLOW MODELING The CVS distributes oxygen to all tissues and organs in the body while removing carbon dioxide. Its three main components are the heart, the blood vessels (arteries, veins, and capillaries), and the blood. Blood that carries carbon dioxide returns to the heart through the veins and is pumped first into the lungs where it releases the CO2 and fills with oxygen. The oxygenated blood returns to the heart and it is then pumped throughout the arteries to distribute the oxygen across the body. Many problems in the CVS are caused by anomalies in the blood vessels; therefore it’s important to understand blood flow trough, mainly, the arterial network. We can distinguish two approaches for modeling blood flow: distributed models and lumped models, Olufsen and Nadim (2004). The first uses fluid dynamics theory to describe the dynamics of blood flow in the arterial system. The second approach relies on simplifications and analogies of the system trying to preserve the main physical interpretation of the components of the systemic circulation. 2.1 Distributed models Blood flow in the arteries is pulsatile, generated by the pumping of the heart and complemented by the elastic properties of the arterial walls. It is a non-Newtonian and non-steady flow. In order to get a detailed and complete mathematical representation of the flow in the circulatory system, the Navier-Stokes equations have to be used. Also, the elastic properties of the arterial wall must be considered to model its effects on the flow, Olufsen and Nadim (2004). These models have the advantage of describing spatial and temporal properties; but they are generally too complicated to use. 2.2 Lumped parameter models These models are easier to understand because, at the cost of omitting the spatial dependence of the variables, the physical interpretation of the parameters is simpler. Windkessel models are the most used lumped models, interpreting the system as an electrical (RCL) circuit. Resistors (R) model the resistance of the arteries to the blood flow, capacitors (C) model the elastic properties of the arterial wall, and inductors (L) account for the inertia of the blood mass. The original Windkessel model included only a resistor and a capacitor in parallel that accounted for the resistance to blood flow and the compliance of the systemic arteries. Later on, a third element, a resistor in series, was added to model the impedance of the aorta, Westerhoff et al. (2009). For instance, the four element Windkessel model is one of the model variations that describe very accurately the pressure in the arteries, Stergiopulos and Westerhof (1999). 3. HYBRID MODEL OF THE CARDIOVASCULAR SYSTEM In Simaan et al. (2009) an electric circuit representation is used to describe the CVS (figure 1). This model is divided into four main parts: 223

Fig. 1. Simplified model of the cardiovascular system, figure extracted from Simaan et al. (2009) • The systemic circulation, modeled as a modified four element Windkessel model. • The left ventricle of the heart, introduced as a time varying capacitance. • The pulmonary circulation and the hearts left atrium are represented as a single capacitance. • The mitral and aortic valves are represented as ideal diodes with a corresponding resistor in series. • A fourth capacitor is introduced to account for the compliance of the big aorta. 3.1 The cardiac cycle The cardiac cycle is divided into four phases depending on the state of the mitral and aortic valves. There are only three phases as one of them repeats. When the pressure in the left atrium (LAP) is higher than the pressure in the left ventricle (LVP), the mitral valve opens allowing the flow of blood into the left ventricle (Filling phase), otherwise the valve closes. At this point both valves are closed and the left ventricle contracts building up pressure (Isovolumic contraction). When pressure in the left ventricle is higher than that in the aorta (AoP), the aortic valve will open and the blood will flow into the arterial system (Ejection phase). As the pressure in the aorta rises above LVP, the aortic valve closes and the left ventricle relaxes (Isovolumic Relaxation). The systemic circulation will be accomplished by means of the pressure of the arterial wall (AP). The wall is expanded when it receives the load of blood and then, by its elastic properties, it returns to its original shape pushing the blood through the system. The behavior of the left ventricular pressure is modeled by a time varying compliance (i.e. capacitance), which is the reciprocal of the elastance. Elastance is a measure of the change in pressure in the left ventricle for a given change in its volume. A mathematical approximation for this function is offered in Simaan et al. (2009). 3.2 Hybrid interpretation The state equations of the system are derived by analysing the circuit using Kirchhoff’s circuit laws and considering the combinations of the states of the mitral and aortic valves, which in turn will derive on different circuits. Table 1 summarizes the operation modes of the system based on the state of the valves. A 0 indicates that the diode is not conducting, i.e the valve is closed. On the other hand, if

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Fig. 2. Hybrid system interpretation of the cardiac cycle the valve is open the diode conducts and is represented by a 1. Table 1. Modes of the cardiac cycle Mitral (DM ) Isovolumic contraction/relaxation Ejection Filling

valve

Aortic (DA )

0

0

0 1

1 0

valve

One can notice that the condition when both valves are open does not exist (non-feasible mode) since it will mean a fatal malfunction. As it can be seen, the system is suitable to be treated as a hybrid system since it has four modes of operation with distinct continuous dynamics. Furthermore, the transition between modes is given by discrete events which is the ideal (nonlinear) operation of the valves (i.e. the diodes). Recall that the opening and closing of the mitral and aortic valves is driven by the difference in pressure between the LAP and LVP and the LVP and AoP correspondingly which are states of the system, i.e. the switching of the modes is state driven. Figure 2 depicts the four modes of the system with the corresponding transition conditions. Let Q = {qi : i ∈ M } with M = {1, 2, 3}, represent the set of possible discrete modes. Define X ⊆ Rn as the continuous time state space of dimension n. The state vector x ∈ Rn belongs to this space. The couple (qi (t), xi (t)) ∈ Q × X represents the state of the hybrid system in mode i. The state equations of the system will be given by, x˙ i = Ai (t)xi (t). (1) The subscript i stands for the active mode qi of the system, i.e. Filling (F ), Ejection (E), or Isovolumic contraction/relaxation (I). The the state vector is defined as, x(t) = [LV P (t) LAP (t) AP (t) AoP (t) QT (t)] . (2) The first four elements of the vector are explained in section 3.1 and QT(t) is the total blood flow in the system. The dynamics matrix, Ai (t), is defined for each of the modes as: Filling phase (q1 = F ):

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  ˙ 1 + C(t)R 1 M − 0 0 0   C(t)RM CRm     1 RM + RS 1  − 0 0    C R Rm CR Rm RS CR RS   1 1 1   A1 (t) =  0 − 0  C S RS CS RS CS     1   0 0 0 0 −  CA   1 1 RC  0 0 − LS LS LS (3) Ejection phase (q2 = E):   ˙ 1 + C(t)R 1 A − 0 0 0   C(t)RA CRA   1 1    0 − 0 0    C R C R R S R S  1 1 1   A2 (t) =  0 − 0   C S RS C S RS CS    1 1 1    0 0 − −  C A RA C A RA C A   1 1 RC  0 0 − LS LS LS (4) Isovolumic phase (q3 = I):  ˙  C(t) − 0 0 0 0  C(t)    1 1    0 − 0 0    C R C R R S R S  1 1 1    (5) A3 (t) =  0 − 0  C S RS C S RS CS    1   0  0 0 0 −  CA   1 1 RC  0 0 − LS LS LS Notice that in each of the modes the A matrix is time varying due to the compliance of the left ventricle.

Let u1 = {0, 0, 1, 0} and u2 = {1, 0, 0, 0} be the natural control inputs sequence of the system given by the mitral and aortic valves correspondingly, i.e. the state of the diodes DM and DA . The values of these signals are discrete and agree with the transitions described in figure 2. By including these inputs in the state space we can define the hybrid representation of the cardiovascular system as in (6).  ˙  C 1 1 − x + (−x + x )u + (−x + x )u 1 2 1 1 4 2  C 1 CRM CRA   1 1 1     − x2 + x3 + (x1 + x2 )   C R RS C R RS CR RM   1 1 1  x(t) ˙ = x − x + x 2 3 5   C S RS CS RS CS     1 1   − x5 + (x1 − x4 )u2   C C R A A A   1 1 RC − x3 + x4 + x5 LS LS LS (6) This model combines both continuous dynamics and discrete events. It provides a mean to analyse possible faults in the discrete transitions of the system. In other words, anomalies in the functioning of the mitral and aortic valves could be studied.

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4. DETECTION OF CARDIOVASCULAR ANOMALIES FDI consists in monitoring a system, identifying when a fault has occurred, and determining the type of fault and its possible cause. There are two approaches for fault detection. In the first one, the pattern of the measured signals is analysed and compared to previously recorded patterns in order to identify abnormal behavior. The second one uses a mathematical model of the system to estimate the behavior of the variables of the system and compare them against those that are measured. FDI methods for hybrid systems exist and have been studied in previous articles, e.g. Cocquempot et al. (2004). In essence, the faults that can occur in a hybrid system are either related with the current mode behavior or the trajectory of the discrete evolution. The model based fault detection method is going to be followed in this section. An extension of the model described in the previous section is used as the basis for the analysis of anomalies in the cardiovascular system. It takes into account the effects of possible anomalies due to variation in the nominal values of the model parameters. 4.1 Cardiovascular model for detection of anomalies The cardiovascular system’s hybrid model described in the previous section depends on some parameters that have physiological meaning. Variation in these parameters from their nominal values can be related to some cardiovascular diseases. For example, consider three common types of CVD’s: atherosclerosis, aneurysm, and high blood pressure. The first two refer, correspondingly, to the hardening of the arteries and the dilatation of the arterial wall. This could be associated with a change in the systemic compliance CS or a variation in the systemic vascular resistance RS . High blood pressure, as its name indicates, describes abnormal levels of systolic and diastolic pressure, NHLBI (2012). The source of this condition could be related to deviations on the normal value of the aortic compliance CA . The presence of this kind of diseases is not always evident, thus, detecting the variations in the parameters could facilitate the diagnosis of abnormal conditions in the CVS. We propose in this section to adapt the cardiovascular hybrid model such that it will be suitable for FDI. Let p be a vector containing the parameters that can suffer a change in its value. Under this condition, the state equation can be expressed in a generic way as x˙ i = fi (xi (t), p, t) (7) In order to account for small variations in the parameter vector p we can do a first order Taylor expansion of (7) to get, dfi (xi (t), p, t) x˙ i ≈ fi (xi (t), p, t) + (p − pN ) (8) dp pN which in turn can be rewritten as: x˙ i = Ai (t)xi (t) + Ei (t)f (t) (9) where f (t) is the faults vector. We have chosen a set of four parameters that can present a variation: the characteristic resistance (RC ), the aortic compliance (CA ), the systemic vascular resistance (RS ), and the systemic compliance 225

(CS ). The components of the fault vector are then the difference with respect to the nominal values of this parameters. Thus, f = [∆RC ∆CA ∆RS ∆CS ] (10) Similarly, the matrix E(t) is the Jacobian matrix of equation (7) with respect to the parameter vector p. This matrix relates the effects of the faults (f ) on the state equation and results from a Taylor expansion of the equations of the model about the nominal values. The resulting three matrices, one per each phase of the cardiac cycle are given by: Filling phase: 

 0 0 0 0   x2 x3 (RM + RS )x2  0  0 − − 0 +   CR RM RS C R RM RS 2 CR RS 2   x x x x x 3 2 2 3 5  0  0 − + − + − 2 2 2 2 2 E1 (t) =  CS RS C S RS CS RS CS RS CS   x   5  0  0 0 2   CA  x5  0 0 0 − LS

(11) Ejection phase:  0 0 0 0 x2 x3  0  0 − 0   C R RS 2 CR RS 2  x2 x3 x2 x3 x5    0 − + − + −  0 2 2 2 2 2 E2 (t) =  CS RS C S RS CS RS CS RS CS    x1 x4 x5  0 −  0 0 2 R + C2 R + C2   CA A A A A  x5  − 0 0 0 LS 

(12) Isovolumic phase:  0 0 0 0 x2 x3  0  0 − 0   CR RS 2 CR RS 2  x2 x3 x2 x3 x5    0 − + − + −  0 2 2 2 2 2 E3 (t) =   C R C R C C C R R S S S S S S S S S   x5  0  0 0 2   C A  x5  − 0 0 0 LS 

(13) Noninvasive AoP and QT measurement Previous studies that aimed to find a lumped parameter model of the cardiovascular system coincide in that invasive measurements of the Aortic Pressure (AoP) and Aortic Flow (QT) were used to determine the parameter of the models, Yu (1998). Pressure: Currently, the main trend in the estimation of the Central Aortic Blood Pressure (CABP) is to indirectly measure it by reconstructing its waveform based on the noninvasive measurement of the blood pressure (BP) in a peripheral artery. The chosen arteries are either the brachial artery or the radial artery. In order to reconstruct the Aortic Pressure waveform data from the peripheral measurement is used together with a Generalized Transfer Function, Williams (2008). Aortic Flow: Regarding the aortic flow, there are a variety of methods that have been in use for the last decade. The most relevant being: • Esophageal Doppler. Measures in real time the cardiac output using transesophageal ultrasonic EchoDoppler probe to measure aortic diameter and aortic flow velocity, Boulnois and Pechoux (2000). • Ultrasound Monitoring. Uses continuous Dopplereffect ultrasound, transducers, algorithms and signal

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processing to measure cardiac output, USCOM Ltd. (2012). • Impedance Cardiography. Monitors cardiac function by measuring pulse-synchrone changes in the thoracic electrical bioimpedance via simple surface electrodes together with a conventional electrocardiogram, Scherhag et al. (2005). Output equation In reality it is not possible to measure all the states of the system. For instance, the three first state variables require invasive methods to be measured. Therefore, an observer is needed in order to estimate the state vector from the available measurements. Considering the information given above, it is assumed here that the aortic pressure (x4 ) and the total flow (x5 ) are available for measurement. Then, the output equation for the system is   00010 yi (t) = Cxi (t), C = . (14) 00001

(a) Pressures

4.2 Residual generation A Luenberger observer, defined as x ˜˙ i (t) = Ai (t)˜ xi (t) + K(t)[yi (t) − C x ˜i (t)], (15) is proposed to estimate xi (t). The observer is used for fault detection by comparing the expected response of the system (9) to the one described by the actual system, estimated by eq. (15). A residual is the result of this comparison and its magnitude is less than a constant value  when there are no faults present. Residuals can be defined in different ways to make the fault suitable for isolation. The simplest residual r(t) is the difference between the output variables: ri (t) = yi (t) − C x ˜i (t) = C (xi (t) − x ˜i (t)) . (16) Define ei = xi − x ˜i as the state estimation error. The time evolution of the error e is defined using (9) and (15) as e˙ i (t) = (Ai (t) − KC)ei + Ei f. (17) When no fault is present, i.e. f = 0, the error will asymptotically converge to the true state. The speed of the convergence is determined by the gain matrix K. If a fault occurs, eq. (17) will not converge to zero due to the forcing term Ei f . In other words, as the observer is not designed to be robust to inputs (i.e. the faults) the estimation will have a variation only caused by the effect of the measured signal (i.e. Ky(t)). Thus, there will be a difference with the actual signal obtained from the model.

(b) Aortic Flow

Fig. 3. MATLAB simulation output

Fig. 4. Comparison between the modeled and observed aortic pressure when a fault occurs

5. SIMULATION

and the shape of the pressure and flow curves agree with those reported in Simaan et al. (2009). The observer was designed to ensure its quick convergence to the fault free system.

The system given in (1) with the dynamics matrices (3) to (5) was simulated using MATLAB and Simulink. The logic followed to determine a change in the phase of the cardiac cycle was that explained in section 3.1. The nominal values used for the parameters are reported in Simaan et al. (2009). Given the periodic nature of the system it is possible to define an initial vector x0 close to the actual values of the states at the start of the simulation. The initial conditions are assumed to be those at the end of the diastolic phase of the cardiac cycle. These values are x0 = [11, 10, 74, 75, 0]T . Figure 3 shows the results of the simulation for the pressures and the flow. Notice that the values are among those considered to be normal

Using eq. (9), we allow the introduction of faults in the model. As given by the fault vector (10), four parameters are considered to be able to vary. In the simulation, these changes in the parameters have been modeled as step changes by some percentage of the nominal value for the parameter. As an example, consider a simulation time of 10 seconds. A fault in the parameter RS of 30% of its nominal value is introduced at t = 5 s. Figure 4 depicts the plot of the modeled and observed aortic pressure. When the fault appears, the modeled aortic pressure increases while the observed counterpart decreases. Similar discrepancies will be encountered in the response of other states variables. Figure 5 presents the evolution of the residual. In the

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REFERENCES

Fig. 5. Errors caused due to the fault (Note: the plot is generated by computing the absolute value of the error divided by the modeled value of the state) first half of the simulation the error is caused by the convergence of the observer as it reaches steady state. When the fault occurs, a particular behavior of increasing and decreasing errors appears. This is a clear indication that a fault has occurred in the system. 6. DISCUSSION AND CONCLUSION In this paper, we have studied the CVS as a hybrid system in order to introduce new techniques for the detection of cardiovascular anomalies. Using the model from Simaan et al. (2009), which can be interpreted as a hybrid system, we proposed an extension of the model considering that anomalies are related to variations in the physiological parameters of the CVS. We assumed that anomalies could be caused by changes in the nominal characteristic resistance, the aortic compliance, the systemic vascular resistance, and the systemic compliance. It was shown how, by looking at the residual generated using an observer, it is possible to detect the occurrence of a fault. However, this is a preliminary study and important work must be done now in the generation of the residual, its analysis, and the interpretation of cardiovascular diseases using variations in the physiological parameters. For instance, a statistical analysis of the residuals seems to be necessary to differentiate between variations that are due to physiological changes and those due to pathological conditions. We believe that modeling the CVS as a hybrid system can be very useful in the detection of anomalies related to diseases in the valves. Many tools are currently developed for fault detection in hybrid system. Our objective is to take advantage of these tools to provide reliable techniques for detection of anomalies. Current work includes the development of residuals that facilitate the isolation of the faults and thus help to identify the cause of the anomaly. Another important aspect, also under consideration, is the appearance of faults in the transitions of the cardiac cycle. The model considers ideal valves with an open or closed state. In reality there could be problems where either the mitral or the aortic valve (or both) are semi opened/closed affecting directly (and severely) the behavior of the state variables, causing problems defining the stage of the cardiac cycle. The hybrid model given in (6) is proposed to explore the set of possible anomalies in the functioning of the mitral and aortic valves. 227

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